Question

the correlation between education and income as measured on 80 people is r = 0.65. explain whether or not each of the following possible conclusions is justified. a) when education increases, income increases as well. b) the form of the relationship between education and income is straight. c) there are no outliers in the scatterplot of income vs. education. d) whether we measure education in years or months, the correlation will still be 0.65. b) is this conclusion justified? explain. a. no, because the absolute value of the correlation coefficient is not equal to 1. b. no, because the correlation coefficient is not equal to 0. c. yes, because the absolute value of the correlation coefficient is equal to 1. d. no, because the form of the relationship cannot be determined from the correlation. e. yes, because the correlation coefficient is close to 1.

Explanation:

Step1: Understand correlation coefficient

The correlation coefficient \(r\) ranges from - 1 to 1. A value of \(r = 0.65\) indicates a moderate - positive linear relationship. A value close to 1 implies a strong positive linear relationship, close to - 1 implies a strong negative linear relationship, and close to 0 implies little to no linear relationship.

Step2: Analyze option a

Just because \(r = 0.65\), we cannot say that when education increases, income increases in a perfectly linear fashion. A correlation coefficient of 1 would be needed for a perfect linear relationship. So, the statement "When Education increases, Income increases as well" is an over - generalization as \(r
eq1\).

Step3: Analyze option b

The form of the relationship (linear or non - linear) cannot be determined solely from the correlation coefficient. The correlation coefficient only measures the strength and direction of the linear relationship. So, we cannot say the form of the relationship between Education and Income is straight just based on \(r = 0.65\).

Step4: Analyze option c

Outliers can affect the correlation coefficient. Just because \(r = 0.65\) does not mean there are no outliers in the scatter - plot of Income vs. Education.

Step5: Analyze option d

Changing the unit of measurement of Education (from years to months) does not change the correlation coefficient. The correlation coefficient is a unitless measure that only depends on the relative positions of the data points in the scatter - plot.

Step6: Analyze option e

Since \(r = 0.65\) which is not close to 1, we cannot say the conclusion that there is a strong linear relationship (implied by a correlation coefficient close to 1) is justified.

Answer:

A. No, because the absolute value of the correlation coefficient is not equal to 1.
B. No, because the form of the relationship cannot be determined from the correlation.
C. No, because there may be outliers affecting the correlation.
D. No, because changing units does not affect the correlation coefficient.
E. No, because the correlation coefficient is not close to 1.